COSE215: Theory of Computation. Lecture 20 P, NP, and NP-Complete Problems

Transcription

1 COSE215: Theory of Computation Lecture 20 P, NP, and NP-Complete Problems Hakjoo Oh 2018 Spring Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

2 Contents 1 P and N P Polynomial-time reductions NP-complete problems 1 The slides are partly based on Siddhartha Sen s slides ( P, NP, and NP-Completeness ) Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

3 Problems Solvable in Polynomial Time (P) A Turing machine M is said to be of time complexity T (n) if whenever M is given an input w of length n, M halts after making at most T (n) moves, regardless of whether or not M accepts. E.g., T (n) = 5n 2, T (n) = 3 n + 5n 4 Polynomial time: T (n) = a 0 n k + a 1 n k a k n + a k+1 We say a language L is in class P if there is some polynomial T (n) such that L = L(M) for some deterministic TM M of time complexity T (n). Problems solvable in polynomial time are called tractable. Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

4 Example: Kruskal s Algorithm A greedy algorithm for finding a minimum-weight spanning tree for a weighted graph. a spanning tree: a subset of the edges such that all nodes are connected through these edges a minimum-weight spanning tree: a spanning tree with the least total weight Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

5 Example: Kruskal s Algorithm Consider the edge (1,3) with the lowest weight (10). Because nodes 1 and 3 are not contained in T at the same time, include the edge in T. Consider the next edge in order of weights: (2,3). Since 2 and 3 are not in T at the same time, include (2,3) in T. Consider the next edge: (1,2). Nodes 1 and 2 are in T. Reject (1,2). Consider the next edge (3,4) and include it in T. We have three edges for the spanning tree of a 4-node graph, so stop. The algorithm takes O(m + e log e) steps (O(n 2 ) for multitape TM). Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

6 Nondeterministic Polynomial Time (N P) We say a language L is in the class N P (nondeterministic polynomial) if there is a nondeterministic TM M and a polynomial time complexity T (n) such that L = L(M), and when M is given an input of length n, there are no sequences of more than T (n) moves of M. Example: TSP (Travelling Salesman Problem) finding a hamiltonian cycle (i.e., a cycle that contains all nodes and each node exactly once) with minimum cost: e.g., To solve TSP, we need to try an exponential number of cycles and compute their total weight. Thus, TSP may not be in P. TSP is in N P because NTM can guess an exponential number of possible solutions and checking a hamiltonian cycle can be done in polynomial time. Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

7 P = N P? One of the deepest open problems. In words: everything that can be done in polynomial time by an NTM can in fact be done by a DTM in polynomial time? P N P because every deterministic TM is a nondeterministic TM. P N P? Probably not. It appears that N P contains many problems not in P. However, no one proved it. Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

8 Implications of P = N P If P=NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in creative leaps, no fundamental gap between solving a problem and recognizing the solution once it s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett. Scott Aaronson Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

9 NP-Complete Problems NP-complete problems are the hardest problems in the NP class. If any NP-complete problem can be solved in polynomial time, then all problems in NP are solvable in polynomial time. How to compare easiness/hardness of problems? Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

10 Problem Solving by Reduction L 1 : the language (problem) to solve L 2 : the problem for which we have an algorithm to solve Solve L 1 by reducing L 1 to L 2 (L 1 L 2 ) via function f: 1 Convert input x of L 1 to instance f(x) of L 2 x L 1 f(x) L 2 2 Apply the algorithm for L 2 to f(x) Running time = time to compute f + time to apply algorithm for L 2 We write L 1 P L 2 if f(x) is computable in polynomial time Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

11 Reductions show easiness/hardness To show L 1 is easy, reduce it to something we know is easy L 1 easy Use algorithm for easy language to decide L 1 To show L 1 is hard, reduce something we know is hard to it (e.g., NP-complete problem) hard L 1 If L 1 was easy, hard would be easy too Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

12 NP-Complete Problems We say L is NP-complete if 1 L is in N P 2 For every language L in N P, there is a polynomial time reduction of L to L (i.e., L P L) Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

13 The Boolean Satisfiability Problem Determine if the given boolean formula can be true. x x x (y z) The first problem proven to be NP-complete. Theorem (Cook-Levin) SAT is NP-complete. We need to show that 1 SAT is NP, and 2 for every L in NP, there is a polynomial-time reduction of L to SAT. Many problems in artificial intelligence, automatic theorem proving, circuit design, etc reduce to SAT/SMT problems. E.g., see Z3 SMT solver ( Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14

14 Summary Undecidable Decidable P N P NP-complete Hakjoo Oh COSE Spring, Lecture 20 June 6, / 14